# Teaching the Slope of a Line and Linear Relationships: What I Learned from Zalman Usiskin

Whenever I attended conferences where the brilliant math educator, Zalman Usiskin, Director of the University of Chicago School Mathematics Project, was giving a presentation, I always made sure I was near the front of the line to get in. Usiskin had a profound influence on how I viewed the world and approached teaching math.

In one of Usiskin’s presentations, he told about a time when he gave his babysitter a ride home. She explained that she did not understand slope of a line. She just didn’t get it. He asked, “ How much per hour did I pay you?” After a few micro-seconds, she came up with the correct answer. To her surprise, he told her that she had just calculated a slope. In the presentation, as Usiskin went on to explain that all slopes are unit rates, I wondered why I hadn’t thought of that before. The amount that the y-variable increases or decreases when the x-variable increases one unit is the slope of the line. After his talk, I placed much more emphasis on teaching slope as rate, such as cost per part, profit per item sold, weight loss per week, etc.

To help teachers with the concept of a slope of a line and linear relationships, you can download my free Linear Growth and Decay handout by going to mathteachersresource.com. The first example in this handout is about predicting gross pay given the number of parts produced in 8 hours. Students seem to have no problem understanding how I derive the equation P(n) = 0.35n + 60. When I ask them why 0.35 in the equation should be no surprise, I’m still amazed that many of them don’t make the connection that the piece work rate stated in the problem was 35 cents per part. But after more coaching, almost all students understand the concept.

Two short examples:

• The first four digits of the square root of 3 = 1.732. What is special about the year 1732?